Dynamic Hedge Ratio for Stock Index Futures: Application of Threshold VECM

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Dynamic Hedge Ratio for Stock Index Futures: Application of Threshold

VECM

Written by Ming-Yuan Leon Li

Department of AccountancyGraduate Institute of Finance and BankingNational Cheng Kung University, Taiwan

July, 2007

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Arbitrage Threshold? From a theoretical point of view, the stock index futures,

in the long run, will eliminate the possibility of arbitrage, equaling the spot index

However, plenty of prior studies announced that the index-futures arbitrageurs only enter into the market if the deviation from the equilibrium relationship is sufficiently large to compensate for transaction costs, as well as risk and price premiums

In other words, for speculators to profit, the difference in the futures and spot prices must be large enough to account the associated costs

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Arbitrage Threshold?

Balke and Formby (1997) serve as one of the first papers to introduce the threshold cointegration model to capture the nonlinear adjustment behaviors of the spot-futures markets.

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Plenty of Prior Studies Yadav et al. (1994), Martens er al. (1998) and Lin,

Cheng and Hwang(2003) for the spot-futures relationship

Anderson (1997) for the yields of T-Bills Michael et al. (1997) and O’Connell (1998) for the

exchange rates Balke and Wohar (1998) for examining interest rate

parity Obstfeld and Taylor (1997), Baum et al. (2001),

Enders and Falk (1998), Lo and Zivot (2001) as well as Taylor (2001) for examining purchasing power parity

Chung et al. (2005) for ADRs.

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Unlike the above Studies…

Adopt a new approach to questions regarding the link between the idea of arbitrage threshold and the establishment of dynamic stock index futures hedge ratio

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Nonlinear Approaches for Hedge Ratio

Bivariate GARCH by Baillie and Myers (1991), Kroner and Sultan (1993), Park and Switzer (1995), Gagnon and Lypny (1995, 1997) and Kavussanos and Nomikos (2000)

Chen et al. (2001) adopted mean-GSV (generalized semi-variance) framework

Miffre (2004) employed conditional OLS approach

Alizadeh and Nomikos (2004) using Markov-switching technique.

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Unlike the above Studies…

Key questions include: Spot and futures prices are more or

less correlated? Volatility/stability of the spot and

futures markets? Design a more efficient hedge ratio? U.S. S&P 500 versus Hungarian BSI

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The Optimal Hedge Ratio

Hedge ratio that minimizes the variance of spot positions:

FF

SSSF

t

tt

FVar

FSCovHR

)(

),(

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Establishing Optimal Hedging Ratio via a No-Threshold System

OLS (Ordinary Least Squares)

VECM (Vector Error Correction Model)

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OLS (Ordinary Least Squares)

OLS (Ordinary Least Squares)

;ttt uFS

HR

FF

SSSF

t

tt

FVar

FSCovHR

)(

),(

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OLS (Ordinary Least Squares)

Weaknesses of OLS Constant variances and correlations Fail to account for the concept of

cointergration

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VECM (Vector Error Correction Model)

VECM (Vector Error Correction Model)

tS

q

jjtjSS

p

iitiSFtSSt

tF

q

jjtjFS

p

iitiFFtFFt

uSFZS

uSFZF

,1

,1

,1

,1

,1

,1

Set up the Zt-1 to be (Ft-1-λ0-λ1 S‧ t-1) which represents the one-period-ahead disequilibrium between futures (Ft-1) and spot (St-1) prices

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VECM (Vector Error Correction Model)

VECM (Vector Error Correction Model)

SS

FF

SF

SF

SS

FF

tS

tF iidu

u

0

0

1

1

0

0,0~

,

,

FF

SSSFHR

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VECM (Vector Error Correction Model)

Weaknesses of VECM Constant variances and correlations Not consider the idea of arbitrage

threshold

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Threshold VECM

Threshold VECM

KtS

q

jjt

KjSS

p

iit

KiSFt

KS

KSt

KtF

q

jjt

KjFS

p

iit

KiFFt

KF

KFt

uSFZS

uSFZF

,1

,1,1

,1,1

,1

,1,1

,1,1

KSS

KFF

KSF

KSF

KSS

KFF

KtS

KtF iid

u

u

0

0

1

1

0

0,0~

,

,

Observable State Variable with Discrete Values: K=1, 2, 3…

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Threshold VECM Threshold VECM with Symmetric

Threshold Parameters

Regime 1 or Central Regime (namely k=1), if |Zt-1| θ≦

Regime 2 or Outer Regime (namely k=2), if |Zt-1|>θ

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Threshold VECM

Regime-varying Hedge Ratio

1

111

KFF

KSSK

SFkHR

2

222

KFF

KSSK

SFkHR

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Threshold VECM The Superiority of Threshold System:

Consider the point of arbitrage threshold Non-constant correlation and volatility A dynamic hedging ratio approach via state-

varying framework Objectively identify the market regime at each

time point (Remember Dummy Variable?) The threshold parameter, namely the θ, could

be estimated by data itself Non-normality problem

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Why Do We Use State-varying Models?

0.00

0.01

0.02

-5 -4.2 -3.5 -2.7 -2 -1.2 -0.5 0.27 1.02 1.77 2.52 3.27 4.02 4.770.00

0.01

0.02

-5 -4.2 -3.4 -2.6 -1.8 -1 -0.2 0.62 1.42 2.22 3.02 3.82 4.62

0.00

0.01

0.02

-5 -4.2 -3.5 -2.7 -2 -1.2 -0.5 0.27 1.02 1.77 2.52 3.27 4.024.77

x11,x12,x13,x14,..

x21

x22

x23

…

x21

x22

x23

x11,x12, .……………… x13,x14

-----Distribution 2: A high Volatility

Distribution

_____Distribution 1: A Low Volatility

Distribution

---- Distribution 2___ Distribution 1

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Data

The daily stock index futures and spot U.S. S&P500 Hungary BSI

January 3. 1996 to December 30, 2005 (2610 observations)

All data is obtained from Datastream database.

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Data

Table 1 Unit Root Tests Cointergration Tests of Stock Index Futures and Spot

U.S. S&P500 Hungarian BSI Futures Spot Future Spot Log levels -2.071 -2.075 -2.975 -3.051 % Returns -13.762* -13.594* -12.080* -11.629* Error Correction Term

-7.4845* -11.1488*

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Data

Table 2 Summary Statistics of Return Rates of Stock Index Futures and Spot

U.S. S&P 500 Hungarian BSI Futures Spots Futures Spots Mean 0.0349 0.0349 0.0972 0.1001 Skewness coefficient

-0.1307 -0.1098 -0.6321 -0.9031

Minimum value -7.7621 -7.1127 -19.678 -18.034 Maximum value 5.7549 5.5732 18.773 13.616 Variance 1.3064 1.1922 4.0349 3.2608 Kurtosis coefficient

6.9938 6.5531 20.263 16.116

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Horse race via a rolling-estimation process

Arbitrage Threshold and Three Key Parameters of Hedge Ratio

Hedging Effectiveness Comparison of Various Alternatives

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Horse race via a rolling-estimation process

Horse races with 1,500-day windows in the rolling estimation process

For each date t, we collect 1,500 pre-daily (t-1 to t-1,500) returns of stock index futures and spot, namely to estimate the parameters of various alternatives

Then we use the parameter estimates of each model to establish the out-sample hedge ratio for date t

500,1

1, iitit FS

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Three Key Parameters for Hedging Ratios

Threshold VECM

KtS

q

jjt

KjSS

p

iit

KiSFt

KS

KSt

KtF

q

jjt

KjFS

p

iit

KiFFt

KF

KFt

uSFZS

uSFZF

,1

,1,1

,1,1

,1

,1,1

,1,1

KSS

KFF

KSF

KSF

KSS

KFF

KtS

KtF iid

u

u

0

0

1

1

0

0,0~

,

,

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Three Key Parameters for Hedging Ratios

Regime 1 or Central Regime (namely k=1), if |Zt-1| θ≦

Regime 2 or Outer Regime (namely k=2), if |Zt-1|>θ

1

111

KFF

KSSK

SFkHR

2

222

KFF

KSSK

SFkHR

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Threshold Parameter Estimates,θ

0

0.002

0.004

0.006

0.008

0.01

0.012

2001/10 2002/2 2002/6 2002/10 2003/2 2003/6 2003/10 2004/2 2004/6 2004/10 2005/2 2005/6 2005/10

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Observation Percentage of Outer

Regime,|Zt-1|>θ

0%

5%

10%

15%

20%

25%

2001/10 2002/2 2002/6 2002/10 2003/2 2003/6 2003/10 2004/2 2004/6 2004/10 2005/2 2005/6 2005/10

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Correlation Coefficient, ρK

S,F

0.9

0.92

0.94

0.96

0.98

1

2001/10 2002/2 2002/6 2002/10 2003/2 2003/6 2003/10 2004/2 2004/6 2004/10 2005/2 2005/6 2005/10

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Standard Error of Futures Position, σK

FF

0.008

0.01

0.012

0.014

0.016

0.018

2001/10 2002/2 2002/6 2002/10 2003/2 2003/6 2003/10 2004/2 2004/6 2004/10 2005/2 2005/6 2005/10

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Standard Error of Spot Position, σK

SS

0.008

0.01

0.012

0.014

0.016

0.018

2001/10 2002/2 2002/6 2002/10 2003/2 2003/6 2003/10 2004/2 2004/6 2004/10 2005/2 2005/6 2005/10

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Relative Standard Error of Spot to Futures, (σK

SS /σK

FF)

0.8

0.85

0.9

0.95

1

2001/10 2002/2 2002/6 2002/10 2003/2 2003/6 2003/10 2004/2 2004/6 2004/10 2005/2 2005/6 2005/10

FF

SSSFHR

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Hedge Ratio Estimates, HR

0.8

0.85

0.9

0.95

1

2001/10 2002/2 2002/6 2002/10 2003/2 2003/6 2003/10 2004/2 2004/6 2004/10 2005/2 2005/6 2005/10

FF

SSSFHR

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Three Key Parameters for HR

U.S. S&P 500 Hungarian BSI

Outer

Regime, k=2

Central

Regime, k=1

Outer

Regime, k=2

Central

Regime, k=1

Correlation Coefficient,

ρkS,F

0.9678 0.9784* 0.5327 0.7238*

Standard Error of Futures

Position, σkFF

0.0140* 0.0129 0.0254* 0.0162

Standard Error of Spot

Position, σkSS

0.0133* 0.0124 0.0229* 0.0171

Relative Standard Error of

Spot to Futures, (σkSS /σ

kFF)

0.9461 0.9637* 0.9539 1.0786*

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Three Key Parameters for HR

The setting without arbitrage threshold will…at the “outer” regime Overestimate the correlation Underestimate the volatility Overestimate the Optimal Hedge Ratio

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Hedging Effectiveness Comparison

For each date t, we use the pre-1,500 daily data to estimate the model parameters and three key parameters of minimum-variance hedge ratio

Next, we establish the minimum-variance hedge ratio for the one-day-after observation

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Hedging Effectiveness Comparison

The variance (namely, Var) of hedged spot position with index futures can be presented as:

)( tt FHRSVar

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Hedging Effectiveness Comparison

Table 4 Hedging Effectiveness of Regime-switching Hedge Ratio via Threshold VECM against Alternative No-threshold Models

U.S. S&P 500 Hungarian BSI

Variance Variance

Reduction

Improvement %

Variance Variance

Reduction

Improvement %

Unhedged 1.141207 - 1.592982 -

OLS 0.043041 96.22848% 0.428891 73.0762%

VECM 0.042324# 96.29133%* 0.34013 78.6482%

Threshold

VECM 0.042629 96.2646% 0.306169# 80.78017%*

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Hedging Effectiveness Comparison

For the case of Hungarian BSI, the threshold systems outperform other alternatives

However, for the case of U.S. S&P 500, the performances of the threshold systems are trivial

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Why???

The θ estimates 0.0066 for U.S. S&P 500 and 0.0322 for

Hungarian BSI 4.8 (=0.0322/0.0066) times A crisis condition versus an unusual

condition

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Why???

Hungarian BSI : HRk=2 is 0.4775 and HRk=1= 0.7825 The difference %=64%

((0.7825-0.4775)/0.4775) U.S. S&P 500

HRk=2 is 0.9158 and HRk=1=0.9430 The difference %=2.96%

((0.9430-0.9158)/0.9158)

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Conclusions The outer regime will be associated with a

smaller correlations, greater volatilities and a smaller value of the optimal hedge ratio

The outer regime as a crisis (unusual) state for the case of Hungarian BSI (U.S. S&P 500)

The superiority of the threshold VECM in enhancing hedging effectiveness especially for the Hungarian BSI market, but not for U.S. S&P 500 market